RESEARCH ARTICLE

Non-convex sparse optimization-based impact force identification with limited vibration measurements

  • Lin CHEN 1,2 ,
  • Yanan WANG , 1,2 ,
  • Baijie QIAO 1,2 ,
  • Junjiang LIU 1,2 ,
  • Wei CHENG 1,2 ,
  • Xuefeng CHEN 1,2
Expand
  • 1. National Key Laboratory of Aerospace Power System and Plasma Technology, Xi’an Jiaotong University, Xi’an 710049, China
  • 2. School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China
yananwang@xjtu.edu.cn

Received date: 28 Feb 2023

Accepted date: 13 Jun 2023

Copyright

2023 Higher Education Press

Abstract

Impact force identification is important for structure health monitoring especially in applications involving composite structures. Different from the traditional direct measurement method, the impact force identification technique is more cost effective and feasible because it only requires a few sensors to capture the system response and infer the information about the applied forces. This technique enables the acquisition of impact locations and time histories of forces, aiding in the rapid assessment of potentially damaged areas and the extent of the damage. As a typical inverse problem, impact force reconstruction and localization is a challenging task, which has led to the development of numerous methods aimed at obtaining stable solutions. The classical 2 regularization method often struggles to generate sparse solutions. When solving the under-determined problem, 2 regularization often identifies false forces in non-loaded regions, interfering with the accurate identification of the true impact locations. The popular 1 sparse regularization, while promoting sparsity, underestimates the amplitude of impact forces, resulting in biased estimations. To alleviate such limitations, a novel non-convex sparse regularization method that uses the non-convex 12 penalty, which is the difference of the 1 and 2 norms, as a regularizer, is proposed in this paper. The principle of alternating direction method of multipliers (ADMM) is introduced to tackle the non-convex model by facilitating the decomposition of the complex original problem into easily solvable subproblems. The proposed method named 12-ADMM is applied to solve the impact force identification problem with unknown force locations, which can realize simultaneous impact localization and time history reconstruction with an under-determined, sparse sensor configuration. Simulations and experiments are performed on a composite plate to verify the identification accuracy and robustness with respect to the noise of the 12-ADMM method. Results indicate that compared with other existing regularization methods, the 12-ADMM method can simultaneously reconstruct and localize impact forces more accurately, facilitating sparser solutions, and yielding more accurate results.

Cite this article

Lin CHEN , Yanan WANG , Baijie QIAO , Junjiang LIU , Wei CHENG , Xuefeng CHEN . Non-convex sparse optimization-based impact force identification with limited vibration measurements[J]. Frontiers of Mechanical Engineering, 2023 , 18(3) : 46 . DOI: 10.1007/s11465-023-0762-2

Nomenclature

Abbreviations
ADMMAlternating direction method of multipliers
BVIDBarely visible impact damage
GREGlobal relative error
IRFImpulse response function
LELocalization error
LRELocal relative error
MC-GSUREMonte Carlo generalized stein unbiased risk estimate
PREPeak relative error
SISOSingle-input single-output
SNRSignal-noise ratio
Variables
a(t)Impulse response function
aij(t)Impulse response function between the output position i and the input location j
aij(ω)Frequency response function between the output position i and the input location j
ATransfer matrix of the multiple-input multiple-output dynamic system
AsTransfer matrix of the single-input single-output dynamic system
BAmplitude of the Gaussian-shaped impact force
cAll-ones vector
CDamping matrix
eRandom noise in measurements
EElastic modulus
fiElements in the vector f
f(t)Impact force excitation function
fForce vector of the multiple-input multiple-output dynamic system
f^Estimated vector of f
fpActual force vector at the impact position p
f^pEstimated force vector at the impact position p
f^MLMaximum likelihood estimation of the vector f
fsForce vector of the single-input single-output dynamic system
g(f)General representation function of penalty terms
GShear modulus
hAn additional vector for variable splitting
h^Estimated vector of h
iLooping variable within the summation operation
IIdentity matrix
kNumber of iterations
KStiffness matrix
mNumber of measurement responses
MMass matrix
nNumber of impact force excitations
nfTotal number of potential impact force locations
NData length of the discretized impulse response function
NmaxMaximum number of iterations
O(nN)Computational complexity of n × N
pSerial number of the location subjected to impact force
qNorm parameter defined in R+
QProjection matrix
rGaussian white noise vector
R(f)General expression for calculating the norm of vector f
s(t)System response
sResponse vector of the multiple-input multiple-output dynamic system
s~Noisy response vector
siResponse vector at a certain position i
ssResponse vector of the single-input single-output dynamic system
tTime
t0Occurrence time instant of the impact
ΔtSampling interval
TImpact duration
uSufficient statistic of the model Eq. (8)
wmaxElement with the largest absolute value in the vector·w
wIntermediate vector defined as w = f (k+1) + z(k)
xλ(u)Solution result of Eq. (8) when f = u
y(f)Proximal operator
δSmall positive parameter
δLagrange multiplier vector
εIteration termination threshold
λRegularization parameter
ρA positive penalty parameter
σStandard deviation of the vector s
σnStandard deviation of noise in the measurements
τTime delayed operator
νPossion’s ratio
ΓThreshold value defined as Γ = λ/ρ

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52075414 and 52241502), and China Postdoctoral Science Foundation (Grant No. 2021M702595).

Conflict of Interest

The authors declare that they have no conflict of interest.
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